and let the parameters of the components in each cluster be the same as those in Sec. REF .
We consider the centralized controller \({\rm col}(u_i)_{i=1}^N=K_{\rm c}{\rm col}(y_i)_{i=1}^N\)  with a dense information structure.
For the centralized controller design, we employ the LQR method under the state weight \(Q_{\rm c}=I_n\otimes \mathrm {D}(q_{\theta },q_{\omega })\)  and the input weight \(R_{\rm c}=10^2I_N\)  with the state observer where \(n\)  is the dimension of all the states.
We also suppose that the global/local subcontrollers are designed in accordance with the previous ones.
The average computation times elapsed for designing the controllers with varied \(n_0\)  ranging from 10 to 25 are depicted in Fig. REF  on a logarithmic scale.
It can be observed that the computation time is significantly reduced through hierarchical model decomposition, which indicates scalability of the proposed distributed design.
Standard model reduction techniques, e.g., balanced truncation method [1]} and singular perturbation method [2]}, are ineffective in this case.
The Hankel singular values \(\sigma _i\)  of the transfer matrix from \({\rm col}(u_i)\)  to \({\rm col}(y_i)\)  for \(n_0=20\)  are given by
\(\sigma _i \in \lbrace 1.3,1.6,1.7,2.4,2.5,+\infty \rbrace ,\quad i=1,\ldots ,360,\) 
